Get the most by viewing this topic in your current grade. Pick your course now.

?
Intros
Lessons
  1. How can we be sure that our general set of solutions is indeed the general set of solutions? The Wronskian!
?
Examples
Lessons
  1. Verifying Some of Our General Solutions
    In the section "Characteristic Equation with Repeated Roots" we had the following differential equation:
    y6y+9y=0y''-6y'+9y=0

    It was found that the two solutions were:
    y1(x)=e3xy_1 (x)=e^{3x}
    y2(x)=xe3xy_2 (x)=xe^{3x}
    1. But one might have made the assumption that both solutions were of the form: y1(x)=y2(x)=e3xy_1 (x)=y_2 (x)=e^{3x}. Demonstrate that these solutions are not linearly independent
    2. Show that the two actual solutions actually form a fundamental set of solutions.
  2. In the section "Characteristic Equation with Real Distinct Roots" we had the following differential equation:
    6y+8y8y=06y''+8y'-8y=0
    We found two solutions:

    y1(x)=e23xy_1 (x)=e^{\frac{2}{3} x}
    y2(x)=e2xy_2 (x)=e^{-2x}

    Verify that these two solutions are indeed linearly independent
    1. A certain differential equation was found to have two solutions:
      y1(x)=3cos(2x)y_1(x)=3 \cos (2x)
      y2(x)=36sin2(x)y_2 (x)=3-6\sin^2 (x)
      Are these two solutions independent? Will they form a fundamental set of solutions?